Character of direct sum of linear representations is sum of characters

Statement

Suppose $G$ is a group, $K$ is a field, and ${\rho }_1:G\to GL(V_1),{\rho }_2:G\to GL(V_2)$ are finite-dimensional linear representations of $G$ over $K$. Denote by ${\chi }_{{\rho }_1},{\chi }_{{\rho }_2}$ the characters of the representations ${\rho }_1,{\rho }_2$ respectively. Denote by ${\rho }_1\oplus {\rho }_2$ the direct sum of linear representations ${\rho }_1,{\rho }_2$ and by ${\chi }_{{\rho }_1\oplus {\rho }_2}$ its character. Then, we have the following for any $g\in G$:

${\chi }_{{\rho }_1\oplus {\rho }_2}={\chi }_{{\rho }_1}(g)+{\chi }_{{\rho }_2}(g)$

Recall that the character of a linear representation is the function that sends any $g$ to the trace of the corresponding linear map.

Related facts

• Character of tensor product of linear representations is product of characters